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Higher Algebra

August 3, 2012

Contents

1 Stable ∞-Categories 131.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.2 The Homotopy Category of a Stable ∞-Category . . . . . . . . . . . . . . . . . . . . . 171.1.3 Closure Properties of Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . 251.1.4 Exact Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2 Stable ∞-Categories and Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.1 t-Structures on Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.2 Filtered Objects and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 371.2.3 The Dold-Kan Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.2.4 The ∞-Categorical Dold-Kan Correspondence . . . . . . . . . . . . . . . . . . . . . . . 52

1.3 Homological Algebra and Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 611.3.1 Nerves of Differential Graded Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 611.3.2 Derived ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.3.3 The Universal Property of D−(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751.3.4 Inverting Quasi-Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831.3.5 Grothendieck Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931.3.6 Complexes of Injective Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

1.4 Spectra and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081.4.1 The Brown Representability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101.4.2 Spectrum Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151.4.3 The ∞-Category of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221.4.4 Presentable Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

2 ∞-Operads 1332.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

2.1.1 From Colored Operads to ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382.1.2 Maps of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432.1.3 Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1482.1.4 ∞-Preoperads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

2.2 Constructions of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542.2.1 Subcategories of O-Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . 1542.2.2 Slicing ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1602.2.3 Coproducts of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1652.2.4 Monoidal Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1682.2.5 Tensor Products of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

2.3 Disintegration and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1782.3.1 Unital ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1792.3.2 Generalized ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

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2.3.3 Approximations to ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1902.3.4 Disintegration of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

2.4 Products and Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2032.4.1 Cartesian Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . . . . . . . . 2042.4.2 Monoid Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2112.4.3 CoCartesian Symmetric Monoidal Structures . . . . . . . . . . . . . . . . . . . . . . . 2132.4.4 Wreath Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

3 Algebras and Modules over ∞-Operads 2273.1 Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

3.1.1 Operadic Colimit Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2283.1.2 Operadic Left Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2383.1.3 Construction of Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2483.1.4 Transitivity of Operadic Left Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . 252

3.2 Limits and Colimits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.2.1 Unit Objects and Trivial Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563.2.2 Limits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2583.2.3 Colimits of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2643.2.4 Tensor Products of Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . 267

3.3 Modules over ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2703.3.1 Coherent ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.3.2 A Coherence Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2753.3.3 Module Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

3.4 General Features of Module ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2903.4.1 Algebra Objects of ∞-Categories of Modules . . . . . . . . . . . . . . . . . . . . . . . 2913.4.2 Modules over Trivial Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3053.4.3 Limits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3083.4.4 Colimits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

4 Associative Algebras and Their Modules 3294.1 Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

4.1.1 The ∞-Operad Ass⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3314.1.2 Simplicial Models for Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 3344.1.3 Monoidal Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3384.1.4 Rectification of Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

4.2 Left and Right Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3524.2.1 The ∞-Operad LM⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3534.2.2 Simplicial Models for Algebras and Modules . . . . . . . . . . . . . . . . . . . . . . . . 3594.2.3 Limits and Colimits of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3674.2.4 Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3714.2.5 Duality in Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

4.3 Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3774.3.1 The ∞-Operad BM⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3784.3.2 Bimodules, Left Modules, and Right Modules . . . . . . . . . . . . . . . . . . . . . . . 3814.3.3 Limits, Colimits, and Free Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3884.3.4 Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3944.3.5 Tensor Products and the Bar Construction . . . . . . . . . . . . . . . . . . . . . . . . 4064.3.6 Associativity of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.3.7 Duality of Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

4.4 Modules over Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4274.4.1 Left and Right Modules over Commutative Algebras . . . . . . . . . . . . . . . . . . . 427

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4.4.2 Tensor Products over Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . 4324.4.3 Change of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4344.4.4 Rectification of Commutative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

5 Little Cubes and Factorizable Sheaves 4455.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

5.1.1 Little Cubes and Configuration Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4485.1.2 The Additivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4535.1.3 Iterated Loop Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4625.1.4 Tensor Products of Ek-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4685.1.5 Coproducts and Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

5.2 Little Cubes and Manifold Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4895.2.1 Embeddings of Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 4895.2.2 Variations on the Little Cubes Operads . . . . . . . . . . . . . . . . . . . . . . . . . . 4945.2.3 Nonunital Ek-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4975.2.4 Little Cubes in a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

5.3 Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5125.3.1 The Ran Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5125.3.2 Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5175.3.3 Properties of Topological Chiral Homology . . . . . . . . . . . . . . . . . . . . . . . . 5225.3.4 Factorizable Cosheaves and Ran Integration . . . . . . . . . . . . . . . . . . . . . . . . 5265.3.5 Verdier Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5325.3.6 Nonabelian Poincare Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

6 Algebraic Structures on ∞-Categories 5496.1 Endomorphism Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

6.1.1 Simplicial Models for Planar ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . 5506.1.2 Endomorphism ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5556.1.3 Nonunital Associative Algebras and their Modules . . . . . . . . . . . . . . . . . . . . 5706.1.4 Deligne’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

6.2 Monads and the Barr-Beck Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5896.2.1 Split Simplicial Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5906.2.2 The Barr-Beck Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5956.2.3 BiCartesian Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6026.2.4 Descent and the Beck-Chevalley Condition . . . . . . . . . . . . . . . . . . . . . . . . 608

6.3 Tensor Products of ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6116.3.1 The Monoidal Structure on Cat∞(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6126.3.2 The Smash Product Monoidal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 6196.3.3 Algebras and their Module Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 6246.3.4 Properties of RModA(C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6296.3.5 Behavior of the Functor Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6396.3.6 Comparison of Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6476.3.7 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

7 The Calculus of Functors 6637.1 The Calculus of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

7.1.1 n-Excisive Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6667.1.2 The Taylor Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6727.1.3 Functors of Many Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6817.1.4 Symmetric Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6897.1.5 Functors from Spaces to Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

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7.1.6 Norm Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7007.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

7.2.1 Derivatives of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7097.2.2 Stabilization of Differentiable Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 7177.2.3 Differentials of Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7267.2.4 Generalized Smash Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7367.2.5 Stabilization of ∞-Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7397.2.6 Uniqueness of Stabilizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

7.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7537.3.1 Cartesian Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7577.3.2 Composition of Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7677.3.3 Derivatives of the Identity Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7737.3.4 Differentiation and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7787.3.5 Consequences of Theorem 7.3.3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7867.3.6 The Dual Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

8 Algebra in the Stable Homotopy Category 8058.1 Structured Ring Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806

8.1.1 E1-Rings and Their Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8078.1.2 Recognition Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8128.1.3 Change of Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8178.1.4 Algebras over Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822

8.2 Properties of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8288.2.1 Free Resolutions and Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 8288.2.2 Flat and Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8358.2.3 Injective Objects of Stable ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 8438.2.4 Localizations and Ore Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8488.2.5 Finiteness Properties of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . 857

8.3 The Cotangent Complex Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8698.3.1 Stable Envelopes and Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 8728.3.2 Relative Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8778.3.3 The Relative Cotangent Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8858.3.4 Tangent Bundles to ∞-Categories of Algebras . . . . . . . . . . . . . . . . . . . . . . . 8948.3.5 The Cotangent Complex of an Ek-Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 9038.3.6 The Tangent Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

8.4 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9118.4.1 Square-Zero Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9128.4.2 Deformation Theory of E∞-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.4.3 Connectivity and Finiteness of the Cotangent Complex . . . . . . . . . . . . . . . . . 930

8.5 Étale Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9418.5.1 Étale Morphisms of E1-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.5.2 The Nonconnective Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9478.5.3 Cocentric Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9538.5.4 Étale Morphisms of Ek-Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958

A Constructible Sheaves and Exit Paths 963A.1 Locally Constant Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964A.2 Homotopy Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967A.3 The Seifert-van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971A.4 Singular Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975A.5 Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977

CONTENTS 7

A.6 ∞-Categories of Exit Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982A.7 A Seifert-van Kampen Theorem for Exit Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 989A.8 Complementary Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994A.9 Exit Paths and Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002

B Categorical Patterns 1011B.1 P-Anodyne Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014B.2 The Model Structure on (Set+∆)/P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023B.3 Flat Inner Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032B.4 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

8 CONTENTS

Let K denote the functor of complex K-theory, which associates to every compact Hausdorff space X theGrothendieck group K(X) of isomorphism classes of complex vector bundles on X. The functor X 7→ K(X)is an example of a cohomology theory: that is, one can define more generally a sequence of abelian groups{Kn(X,Y )}n∈Z for every inclusion of topological spaces Y ⊆ X, in such a way that the Eilenberg-Steenrodaxioms are satisfied (see [49]). However, the functor K is endowed with even more structure: for everytopological space X, the abelian group K(X) has the structure of a commutative ring (when X is compact,the multiplication on K(X) is induced by the operation of tensor product of complex vector bundles). Onewould like that the ring structure on K(X) is a reflection of the fact that K itself has a ring structure, in asuitable setting.

To analyze the problem in greater detail, we observe that the functor X 7→ K(X) is representable. Thatis, there exists a topological space Z = Z×BU and a universal class η ∈ K(Z), such that for every sufficientlynice topological space X, the pullback of η induces a bijection [X,Z] → K(X); here [X,Z] denotes the setof homotopy classes of maps from X into Z. According to Yoneda’s lemma, this property determines thespace Z up to homotopy equivalence. Moreover, since the functor X 7→ K(X) takes values in the categoryof commutative rings, the topological space Z is automatically a commutative ring object in the homotopycategory H of topological spaces. That is, there exist addition and multiplication maps Z × Z → Z, suchthat all of the usual ring axioms are satisfied up to homotopy. Unfortunately, this observation is not veryuseful. We would like to have a robust generalization of classical algebra which includes a good theory ofmodules, constructions like localization and completion, and so forth. The homotopy category H is toopoorly behaved to support such a theory.

An alternate possibility is to work with commutative ring objects in the category of topological spacesitself: that is, to require the ring axioms to hold “on the nose” and not just up to homotopy. Althoughthis does lead to a reasonable generalization of classical commutative algebra, it not sufficiently general formany purposes. For example, if Z is a topological commutative ring, then one can always extend the functorX 7→ [X,Z] to a cohomology theory. However, this cohomology theory is not very interesting: in degreezero, it simply gives the following variant of classical cohomology:∏

n≥0

Hn(X;πnZ).

In particular, complex K-theory cannot be obtained in this way. In other words, the Z = Z×BU for stablevector bundles cannot be equipped with the structure of a topological commutative ring. This reflects thefact that complex vector bundles on a space X form a category, rather than just a set. The direct sum andtensor product operation on complex vector bundles satisfy the ring axioms, such as the distributive law

E⊗(F⊕F′) ' (E⊗F)⊕ (E⊗F′),

but only up to isomorphism. However, although Z× BU has less structure than a commutative ring, it hasmore structure than simply a commutative ring object in the homotopy category H, because the isomorphismdisplayed above is actually canonical and satisfies certain coherence conditions (see [91] for a discussion).

To describe the kind of structure which exists on the topological space Z × BU, it is convenient tointroduce the language of commutative ring spectra, or, as we will call them, E∞-rings. Roughly speaking,an E∞-ring can be thought of as a space Z which is equipped with an addition and a multiplication forwhich the axioms for a commutative ring hold not only up to homotopy, but up to coherent homotopy.The E∞-rings play a role in stable homotopy theory analogous to the role played by commutative rings inordinary algebra. As such, they are the fundamental building blocks of derived algebraic geometry.

One of our ultimate goals in this book is to give an exposition of the theory of E∞-rings. Recall thatordinary commutative ring R can be viewed as a commutative algebra object in the category of abeliangroups, which we view as endowed with a symmetric monoidal structure given by tensor product of abeliangroups. To obtain the theory of E∞-rings we will use the same definition, replacing abelian groups by spectra(certain algebro-topological objects which represent cohomology theories). To carry this out in detail, weneed to say exactly what a spectrum is. There are many different definitions in the literature, having a

CONTENTS 9

variety of technical advantages and disadvantages. Some modern approaches to stable homotopy theoryhave the feature that the collection of spectra is realized as a symmetric monoidal category (and one candefine an E∞-ring to be a commutative algebra object of this category): see, for example, [73].

We will take a different approach, using the framework of ∞-categories developed in [97]. The collectionof all spectra can be organized into an ∞-category, which we will denote by Sp: it is an ∞-categoricalcounterpart of the ordinary category of abelian groups. The tensor product of abelian groups also has acounterpart: the smash product functor on spectra. In order to describe the situation systematically, weintroduce the notion of a symmetric monoidal ∞-category: that is, an ∞-category C equipped with a tensorproduct functor ⊗ : C×C → C which is commutative and associative up to coherent homotopy. For anysymmetric monoidal ∞-category C, there is an associated theory of commutative algebra objects, which arethemselves organized into an ∞-category CAlg(C). We can then define an E∞-ring to be a commutativealgebra object of the∞-category of spectra, endowed with the symmetric monoidal structure given by smashproducts.

We now briefly outline the contents of this book (more detailed outlines can be found at the beginningof individual sections and chapters). Much of this book is devoted to developing an adequate language tomake sense of the preceding paragraph. We will begin in Chapter 1 by introducing the notion of a stable∞-category. Roughly speaking, the notion of stable ∞-category is obtained by axiomatizing the essentialfeature of stable homotopy theory: fiber sequences are the same as cofiber sequences. The ∞-category Spof spectra is an example of a stable ∞-category. In fact, it is universal among stable ∞-categories: we willshow that Sp is freely generated (as a stable ∞-category which admits small colimits) by a single object(see Corollary 1.4.4.6). However, there are a number of stable ∞-categories that are of interest in othercontexts. For example, the derived category of an abelian category can be realized as the homotopy categoryof a stable ∞-category. We may therefore regard the theory of stable ∞-categories as a generalization ofhomological algebra, which has many applications in pure algebra and algebraic geometry.

We can think of an ∞-category C as comprised of a collection of objects X,Y, Z, . . . ∈ C, togetherwith a mapping space MapC(X,Y ) for every pair of objects X,Y ∈ C (which are equipped with coherentlyassociative composition laws). In Chapter 2, we will study a variation on the notion of∞-category, which wecall an∞-operad. Roughly speaking, an∞-operad O consists of a collection of objects together with a spaceof operations MulO({Xi}1≤i≤n, Y )} for every finite collection of objects X1, . . . , Xn, Y ∈ O (again equippedwith coherently associative multiplication laws). As a special case, we will obtain a theory of symmetricmonoidal ∞-categories.

Given a pair of ∞-operads O and C, the collection of maps from O to C is naturally organized into an∞-category which we will denote by AlgO(C), and refer to as the ∞-category of O-algebra objects of C. Animportant special case is when O is the commutative ∞-operad and C is a symmetric monoidal ∞-category:in this case, we will refer to AlgO(C) as the ∞-category of commutative algebra objects of C and denote it byCAlg(C). We will make a thorough study of algebra objects (commutative and otherwise) in Chapter 3.

In Chapter 4, we will specialize our general theory of algebras to the case where O is the associative ∞-operad. In this case, we will denote AlgO(C) by Alg(C) and refer to it the ∞-category of associative algebraobjects of C. The ∞-categorical theory of associative algebra objects is an excellent formal parallel of theusual theory of associative algebras. For example, one can study left modules, right modules, and bimodulesover associative algebras. This theory of modules has some nontrivial applications. For example, we willuse it in Chapter 6 to prove a version of Deligne’s conjecture (regarding the structure of the Hochschildcochain complex of an associative algebra) and an ∞-categorical analogue of the Barr-Beck theorem, whichhas many applications in higher category theory.

In ordinary algebra, there is a thin line dividing the theory of commutative rings from the theory ofassociative rings: a commutative ring R is just an associative ring whose elements satisfy the additionalidentity xy = yx. In the ∞-categorical setting, the situation is rather different. Between the theory ofassociative and commutative algebras is a whole hierarchy of intermediate notions of commutativity, whichare described by the “little cubes” operads of Boardman and Vogt. In Chapter 5, we will introduce thenotion of an Ek-algebra for each 0 ≤ k ≤ ∞. This definition reduces to the notion of an associative algebrain the case k = 1, and to the notion of a commutative algebra when k =∞. The theory of Ek-algebras has

10 CONTENTS

many applications in intermediate cases 1 < k

CONTENTS 11

• We will make extensive use of definitions and notations from the book [97]. If the reader encounterssomething confusing or unfamiliar, we recommend looking there first. We adopt the convention thatreferences to [97] will be indicated by use of the letter T. For example, Theorem T.6.1.0.6 refers toTheorem 6.1.0.6 of [97].

• We say that a category (or ∞-category) C is presentable if C admits small colimits and is generatedunder small colimits by a set of κ-compact objects, for some regular cardinal number κ. This isdeparture from the standard category-theoretic terminology, in which such categories are called locallypresentable (see [1]).

• We let Set∆ denote the category of simplicial sets. If J is a linearly ordered set, we let ∆J denotethe simplicial set given by the nerve of J , so that the collection of n-simplices of ∆J can be identifiedwith the collection of all nondecreasing maps {0, . . . , n} → J . We will frequently apply this notationwhen J is a subset of {0, . . . , n}; in this case, we can identify ∆J with a subsimplex of the standardn-simplex ∆n (at least if J 6= ∅; if J = ∅, then ∆J is empty).

• We will often use the term space to refer to a Kan complex (that is, a simplicial set satisfying the Kanextension condition).

• Let n ≥ 0. We will say that a space X is n-connective if it is nonempty and the homotopy setsπi(X,x) are trivial for i < n and every vertex x of X (spaces with this property are more commonlyreferred to as (n− 1)-connected in the literature). We say that X is connected if it is 1-connective. Byconvention, we say that every space X is (−1)-connective. We will say that a map of spaces f : X → Yis n-connective if the homotopy fibers of f are n-connective.

• Let n ≥ −1. We say that a space X is n-truncated if the homotopy sets πi(X,x) are trivial for everyi > n and every vertex x ∈ X. We say that X is discrete if it is 0-truncated. By convention, we saythat X is (−2)-truncated if and only if X is contractible. We will say that a map of spaces f : X → Yis n-truncated if the homotopy fibers of f are n-truncated.

• Throughout this book, we will use homological indexing conventions whenever we discuss homologicalalgebra. For example, chain complexes of abelian groups will be denoted by

· · · → A2 → A1 → A0 → A−1 → A−2 → · · · ,

with the differential lowering the degree by 1.

• In Chapter 1, we will construct an ∞-category Sp, whose homotopy category hSp can be identifiedwith the classical stable homotopy category. In Chapter 8, we will construct a symmetric monoidalstructure on Sp, which gives (in particular) a tensor product functor Sp×Sp→ Sp. At the level of thehomotopy category hSp, this functor is given by the classical smash product of spectra, which is usuallydenoted by (X,Y ) 7→ X ∧ Y . We will adopt a different convention, and denote the smash productfunctor by (X,Y ) 7→ X ⊗ Y .

• If A is a model category, we let Ao denote the full subcategory of A spanned by the fibrant-cofibrantobjects.

• Let C be an ∞-category. We let C' denote the largest Kan complex contained in C: that is, the∞-category obtained from C by discarding all noninvertible morphisms.

• Let C be an ∞-category containing objects X and Y . We let CX/ and C/Y denote the undercategoryand overcategory defined in §T.1.2.9. We will generally abuse notation by identifying objects of these∞-categories with their images in C. If we are given a morphism f : X → Y , we can identify X withan object of C/Y and Y with an object of CX/, so that the ∞-categories

(CX/)/Y (C/Y )X/

12 CONTENTS

are defined (and canonically isomorphic as simplicial sets). We will denote these ∞-categories byCX//Y (beware that this notation is slightly abusive: the definition of CX//Y depends not only on C,X, and Y , but also on the morphism f).

• Let C and D be ∞-categories. We let FunL(C,D) denote the full subcategory of Fun(C,D) spanned bythose functors which admit right adjoints, and FunR(C,D) the full subcategory of Fun(C,D) spannedby those functors which admit left adjoints. If C and D are presentable, then these subcategories admita simpler characterization: a functor F : C→ D belongs to FunL(C,D) if and only if it preserves smallcolimits, and belongs to FunR(C,D) if and only if it preserves small limits and small κ-filtered colimitsfor a sufficiently large regular cardinal κ (see Corollary T.5.5.2.9).

• We will say that a map of simplicial sets f : S → T is left cofinal if, for every right fibration X → T ,the induced map of simplicial sets FunT (T,X) → FunT (S,X) is a homotopy equivalence of Kancomplexes (in [97], we referred to a map with this property as cofinal). We will say that f is rightcofinal if the induced map Sop → T op is left cofinal: that is, if f induces a homotopy equivalenceFunT (T,X)→ FunT (S,X) for every left fibration X → T . If S and T are ∞-categories, then f is leftcofinal if and only if for every object t ∈ T , the fiber product S×T Tt/ is weakly contractible (TheoremT.4.1.3.1).

Acknowledgements

In writing this book, I have benefited from the advice and assistance of many people. I would like to thank BenAntieau, Tobias Barthel, Dario Beraldo, Daniel Brügmann, Moritz Groth, Rune Haugseng, Vladimir Hinich,Allen Knutson, Joseph Lipman, Akhil Mathew, Anatoly Preygel, Steffen Sagave, Christian Schlichtkrull,Timo Schürg, Markus Spitzweck, Hiro Tanaka, Arnav Tripathy, and James Wallbridge for locating manymistakes in earlier versions of this book (though I am sure that there are many left to find). I would also liketo thank Matt Ando, Clark Barwick, David Ben-Zvi, Alexander Beilinson, Julie Bergner, Andrew Blumberg,Dustin Clausen, Dan Dugger, Vladimir Drinfeld, Matt Emerton, John Francis, Dennis Gaitsgory, AndreHenriques, Gijs Heuts, Mike Hopkins, Andre Joyal, Ieke Moerdijk, David Nadler, Anatoly Preygel, CharlesRezk, David Spivak, Bertrand Toën, and Gabriele Vezzosi for useful conversations related to the subjectmatter of this book. Finally, I would like to thank the National Science Foundation for supporting thisproject under grant number 0906194.

Chapter 1

Stable ∞-Categories

There is a very useful analogy between topological spaces and chain complexes with values in an abeliancategory. For example, it is customary to speak of homotopies between chain maps, contractible complexes,and so forth. The analogue of the homotopy category of topological spaces is the derived category of an abeliancategory A, a triangulated category which provides a good setting for many constructions in homologicalalgebra. However, it has long been recognized that for many purposes the derived category is too crude:it identifies homotopic morphisms of chain complexes without remembering why they are homotopic. It ispossible to correct this defect by viewing the derived category as the homotopy category of an underlying∞-category D(A). The∞-categories which arise in this way have special features that reflect their “additive”origins: they are stable.

We will begin in §1.1 by giving the definition of stability and exploring some of its consequences. Forexample, we will show that if C is a stable ∞-category, then its homotopy category hC is triangulated(Theorem 1.1.2.14), and that stable ∞-categories admit finite limits and colimits (Proposition 1.1.3.4). Theappropriate notion of functor between stable ∞-categories is an exact functor: that is, a functor whichpreserves finite colimits (or equivalently, finite limits: see Proposition 1.1.4.1). The collection of stable ∞-categories and exact functors between them can be organized into an ∞-category, which we will denote byCatEx∞ . In §1.1.4, we will establish some basic closure properties of the ∞-category Cat

Ex∞ ; in particular, we

will show that it is closed under the formation of limits and filtered colimits in Cat∞. The formation oflimits in CatEx∞ provides a tool for addressing the classical problem of “gluing in the derived category”.

In §1.2, we recall the definition of a t-structure on a triangulated category. If C is a stable ∞-category,we define a t-structure on C to be a t-structure on its homotopy category hC. If C is equipped with at-structure, we show that every filtered object of C gives rise to a spectral sequence taking values in theheart C♥ (Proposition 1.2.2.7). In particular, we show that every simplicial object of C determines a spectralsequence, using an ∞-categorical analogue of the Dold-Kan correspondence.

We will return to the setting of homological algebra in §1.3. To any abelian category A with enoughprojective objects, one can associate a stable ∞-category D−(A), whose objects are (right-bounded) chaincomplexes of projective objects of A. This ∞-category provides useful tools for organizing informationin homological algebra. Our main result (Theorem 1.3.3.8) is a characterization of D−(A) by a universalmapping property.

In §1.4, we will focus our attention on a particular stable ∞-category: the ∞-category Sp of spectra.The homotopy category of Sp can be identified with the classical stable homotopy category, which is thenatural setting for a large portion of modern algebraic topology. Roughly speaking, a spectrum is a sequenceof pointed spaces {X(n)}n∈Z equipped with homotopy equivalences X(n) ' ΩX(n + 1), where Ω denotesthe functor given by passage to the loop space. More generally, one can obtain a stable ∞-category byconsidering sequences as above which take values in an arbitrary ∞-category C which admits finite limits;we denote this ∞-category by Sp(C) and refer to it as the ∞-category of spectrum objects of C.

13

14 CHAPTER 1. STABLE ∞-CATEGORIES

1.1 Foundations

Our goal in this section is to introduce our main object of study for this chapter: the notion of a stable ∞-category. The theory of stable ∞-categories can be regarded as an axiomatization of the essential featuresof stable homotopy theory: most notably, that fiber sequences and cofiber sequences are the same. Wewill begin in §1.1.1 by reviewing some of the relevant notions (pointed ∞-categories, zero objects, fiber andcofiber sequences) and using them to define the class of stable ∞-categories.

In §1.1.2, we will review Verdier’s definition of a triangulated category. We will show that if C is a stable∞-category, then its homotopy category hC has the structure of a triangulated category (Theorem 1.1.2.14).The theory of triangulated categories can be regarded as an attempt to capture those features of stable∞-categories which are easily visible at the level of homotopy categories. Triangulated categories whicharise naturally in mathematics are usually given as the homotopy categories of stable ∞-categories, thoughit is possible to construct triangulated categories which are not of this form (see [113]).

Our next goal is to study the properties of stable ∞-categories in greater depth. In §1.1.3, we will showthat a stable ∞-category C admits all finite limits and colimits, and that pullback squares and pushoutsquares in C are the same (Proposition 1.1.3.4). We will also show that the class of stable ∞-categories isclosed under various natural operations. For example, we will show that if C is a stable ∞-category, thenthe ∞-category of Ind-objects Ind(C) is stable (Proposition 1.1.3.6), and that the ∞-category of diagramsFun(K,C) is stable for any simplicial set K (Proposition 1.1.3.1).

In §1.1.4, we shift our focus somewhat. Rather than concerning ourselves with the properties of anindividual stable ∞-category C, we will study the collection of all stable ∞-categories. To this end, weintroduce the notion of an exact functor between stable ∞-categories. We will show that the collection ofall (small) stable ∞-categories and exact functors between them can itself be organized into an ∞-categoryCatEx∞ , and study some of the properties of Cat

Ex∞ .

Remark 1.1.0.1. The theory of stable ∞-categories is not really new: most of the results presented hereare well-known to experts. There exists a growing literature on the subject in the setting of stable modelcategories: see, for example, [37], [126], [128], and [72]. For a brief account in the more flexible setting ofSegal categories, we refer the reader to [152].

Remark 1.1.0.2. Let k be a field. Recall that a differential graded category over k is a category enrichedover the category of chain complexes of k-vector spaces. The theory of differential graded categories is closelyrelated to the theory of stable∞-categories. More precisely, one can show that the data of a (pretriangulated)differential graded category over k is equivalent to the data of a stable ∞-category C equipped with anenrichment over the monoidal ∞-category of k-module spectra. The theory of differential graded categoriesprovides a convenient language for working with stable ∞-categories of algebraic origin (for example, thosewhich arise from chain complexes of coherent sheaves on algebraic varieties), but is inadequate for treatingexamples which arise in stable homotopy theory. There is a voluminous literature on the subject; see, forexample, [84], [101], [141], [35], and [147].

1.1.1 Stability

In this section, we introduce the definition of a stable ∞-category. We begin by reviewing some definitionsfrom [97].

Definition 1.1.1.1. Let C be an ∞-category. A zero object of C is an object which is both initial and final.We will say that C is pointed if it contains a zero object.

In other words, an object 0 ∈ C is zero if the spaces MapC(X, 0) and MapC(0, X) are both contractiblefor every object X ∈ C. Note that if C contains a zero object, then that object is determined up toequivalence. More precisely, the full subcategory of C spanned by the zero objects is a contractible Kancomplex (Proposition T.1.2.12.9).

1.1. FOUNDATIONS 15

Remark 1.1.1.2. Let C be an ∞-category. Then C is pointed if and only if the following conditions aresatisfied:

(1) The ∞-category C has an initial object ∅.

(2) The ∞-category C has a final object 1.

(3) There exists a morphism f : 1→ ∅ in C.

The “only if” direction is obvious. For the converse, let us suppose that (1), (2), and (3) are satisfied. Weinvoke the assumption that ∅ is initial to deduce the existence of a morphism g : ∅ → 1. Because ∅ is initial,f ◦g ' id∅, and because 1 is final, g ◦f ' id1. Thus g is a homotopy inverse to f , so that f is an equivalence.It follows that ∅ is also a final object of C, so that C is pointed.

Remark 1.1.1.3. Let C be an ∞-category with a zero object 0. For any X,Y ∈ C, the natural map

MapC(X, 0)×MapC(0, Y )→ MapC(X,Y )

has contractible domain. We therefore obtain a well defined morphism X → Y in the homotopy categoryhC, which we will refer to as the zero morphism and also denote by 0.

Definition 1.1.1.4. Let C be a pointed∞-category. A triangle in C is a diagram ∆1×∆1 → C, depicted as

Xf //

��

Y

g

��0 // Z

where 0 is a zero object of C. We will say that a triangle in C is a fiber sequence if it is a pullback square,and a cofiber sequence if it is a pushout square.

Remark 1.1.1.5. Let C be a pointed ∞-category. A triangle in C consists of the following data:

(1) A pair of morphisms f : X → Y and g : Y → Z in C.

(2) A 2-simplex in C corresponding to a diagram

Yg

��X

f>>

h // Z

in C, which identifies h with the composition g ◦ f .

(3) A 2-simplex0

��X

??

h // Z

in C, which we may view as a nullhomotopy of h.

We will generally indicate a triangle by specifying only the pair of maps

Xf→ Y g→ Z,

with the data of (2) and (3) being implicitly assumed.


Definition 1.1.1.6. Let C be a pointed ∞-category containing a morphism g : X → Y . A fiber of g is afiber sequence

W //

��

X

g

��0 // Y.

Dually, a cofiber of g is a cofiber sequence

Xg //

��

Y

��0 // Z.

We will generally abuse terminology by simply referring to W and Z as the fiber and cofiber of g. We willalso write W = fib(g) and Z = cofib(g).

Remark 1.1.1.7. Let C be a pointed ∞-category containing a morphism f : X → Y . A cofiber off , if it exists, is uniquely determined up to equivalence. More precisely, consider the full subcategoryE ⊆ Fun(∆1 × ∆1,C) spanned by the cofiber sequences. Let θ : E → Fun(∆1,C) be the forgetful functor,which associates to a diagram

Xg //

��

Y

��0 // Z

the morphism g : X → Y . Applying Proposition T.4.3.2.15 twice, we deduce that θ is a Kan fibration, whosefibers are either empty or contractible (depending on whether or not a morphism g : X → Y in C admitsa cofiber). In particular, if every morphism in C admits a cofiber, then θ is a trivial Kan fibration, andtherefore admits a section cofib : Fun(∆1,C)→ Fun(∆1 ×∆1,C), which is well defined up to a contractiblespace of choices. We will often abuse notation by also letting cofib : Fun(∆1,C)→ C denote the composition

Fun(∆1,C)→ Fun(∆1 ×∆1,C)→ C,

where the second map is given by evaluation at the final object of ∆1 ×∆1.

Remark 1.1.1.8. The functor cofib : Fun(∆1,C) → C can be identified with a left adjoint to the left Kanextension functor C ' Fun({1},C) → Fun(∆1,C), which associates to each object X ∈ C a zero morphism0→ X. It follows that cofib preserves all colimits which exist in Fun(∆1,C) (Proposition T.5.2.3.5).

Definition 1.1.1.9. An ∞-category C is stable if it satisfies the following conditions:

(1) There exists a zero object 0 ∈ C.

(2) Every morphism in C admits a fiber and a cofiber.

(3) A triangle in C is a fiber sequence if and only if it a cofiber sequence.

Remark 1.1.1.10. Condition (3) of Definition 1.1.1.9 is analogous to the axiom for abelian categories whichrequires that the image of a morphism be isomorphic to its coimage.

Example 1.1.1.11. Recall that a spectrum consists of an infinite sequence of pointed topological spaces{Xi}i≥0, together with homeomorphisms Xi ' ΩXi+1, where Ω denotes the loop space functor. Thecollection of spectra can be organized into a stable ∞-category Sp. Moreover, Sp is in some sense theuniversal example of a stable ∞-category. This motivates the terminology of Definition 1.1.1.9: an ∞-category C is stable if it resembles the ∞-category Sp, whose homotopy category hSp can be identified withthe classical stable homotopy category. We will return to the theory of spectra (using a slightly differentdefinition) in §1.4.3.

1.1. FOUNDATIONS 17

Example 1.1.1.12. Let A be an abelian category. Under mild hypotheses, we can construct a stable ∞-category D(A) whose homotopy category hD(A) can be identified with the derived category of A, in thesense of classical homological algebra. We will outline the construction of D(A) in §1.3.2.

Remark 1.1.1.13. If C is a stable ∞-category, then the opposite ∞-category Cop is also stable.

Remark 1.1.1.14. One attractive feature of the theory of stable∞-categories is that stability is a propertyof ∞-categories, rather than additional data. The situation for additive categories is similar. Althoughadditive categories are often presented as categories equipped with additional structure (an abelian groupstructure on all Hom-sets), this additional structure is in fact determined by the underlying category: seeDefinition 1.1.2.1. The situation for stable ∞-categories is similar: we will see later that every stable ∞-category is canonically enriched over the ∞-category of spectra.

1.1.2 The Homotopy Category of a Stable ∞-CategoryLet M be a module over a commutative ring R. Then M admits a resolution

· · · → P2 → P1 → P0 →M → 0

by projective R-modules. In fact, there are generally many choices for such a resolution. Two projectiveresolutions of M need not be isomorphic to one another. However, they are always quasi-isomorphic: that is,if we are given two projective resolutions P• and P

′• of M , then there is a map of chain complexes P• → P ′•

which induces an isomorphism on homology groups. This phenomenon is ubiquitous in homological algebra:many constructions produce chain complexes which are not really well-defined up to isomorphism, but only upto quasi-isomorphism. In studying these constructions, it is often convenient to work in the derived categoryD(R) of the ring R: that is, the category obtained from the category of chain complexes of R-modules byformally inverting all quasi-isomorphisms.

The derived category D(R) of a commutative ring R is usually not an abelian category. For example, amorphism f : X ′ → X in D(R) usually does not have a cokernel in D(R). Instead, one can associate to fits cofiber (or mapping cone) X ′′, which is well-defined up to noncanonical isomorphism. In [154], Verdierintroduced the notion of a triangulated category in order to axiomatize the structure on D(R) given bythe formation of mapping cones. In this section, we will review Verdier’s theory of triangulated categories(Definition 1.1.2.5) and show that the homotopy category of a stable∞-category C is triangulated (Theorem1.1.2.14).

We begin with some basic definitions.

Definition 1.1.2.1. Let A be a category. We will say that A is additive if it satisfies the following fourconditions:

(1) The category A admits finite products and coproducts.

(2) The category A has a zero object, which we will denote by 0.

For any pair of objects X,Y ∈ A, a zero morphism from X to Y is a map f : X → Y which factors as acomposition X → 0→ Y . It follows from (2) that for every pair X,Y ∈ A, there is a unique zero morphismfrom X to Y , which we will denote by 0.

(3) For every pair of objects X,Y , the map X∐Y → X × Y described by the matrix[idX 00 idY

]is an isomorphism; let φX,Y denote its inverse.


Assuming (3), we can define the sum of two morphisms f, g : X → Y to be the morphism f + g given by thecomposition

X → X ×X f,g→ Y × Y φY,Y→ Y∐

Y → Y.

It is easy to see that this construction endows HomA(X,Y ) with the structure of a commutative monoid,whose identity is the unique zero morphism from X to Y .

(4) For every pair of objects X,Y ∈ A, the addition defined above determines a group structure onHomA(X,Y ). In other words, for every morphism f : X → Y , there exists another morphism −f :X → Y such that f + (−f) is a zero morphism from X to Y .

Remark 1.1.2.2. An additive category A is said to be abelian if every morphism f : X → Y in A admits akernel and a cokernel, and the canonical map coker(ker(f)→ X)→ ker(Y → coker(f)) is an isomorphism.

Remark 1.1.2.3. Let A be an additive category. Then the composition law on A is bilinear: for pairs ofmorphisms f, f ′ ∈ HomA(X,Y ) and g, g′ ∈ HomA(Y,Z), we have

g ◦ (f + f ′) = (g ◦ f) + (g ◦ f ′) (g + g′) ◦ f = (g ◦ f) + (g′ ◦ f).

In other words, the composition law on A determines abelian group homomorphisms

HomA(X,Y )⊗HomA(Y,Z)→ HomA(X,Z).

We can summarize the situation by saying that the category A is enriched over the category of abeliangroups.

Remark 1.1.2.4. Let A be an additive category. It follows from condition (3) of Definition 1.1.2.1 that forevery pair of objects X,Y ∈ A, the product X × Y is canonically isomorphic to the coproduct X

∐Y . It is

customary to emphasize this identification by denoting both the product and the coproduct by X ⊕ Y ; wewill refer to X ⊕ Y as the direct sum of X and Y .

Definition 1.1.2.5 (Verdier). A triangulated category consists of the following data:

(1) An additive category D.

(2) A translation functor D → D which is an equivalence of categories. We denote this functor byX 7→ X[1].

(3) A collection of distinguished triangles

Xf→ Y g→ Z h→ X[1].

These data are required to satisfy the following axioms:

(TR1) (a) Every morphism f : X → Y in D can be extended to a distinguished triangle in D.(b) The collection of distinguished triangles is stable under isomorphism.

(c) Given an object X ∈ D, the diagram

XidX→ X → 0→ X[1]

is a distinguished triangle.

(TR2) A diagram

Xf→ Y g→ Z h→ X[1]

is a distinguished triangle if and only if the rotated diagram

Yg→ Z h→ X[1] −f [1]→ Y [1]

is a distinguished triangle.

1.1. FOUNDATIONS 19

(TR3) Given a commutative diagram

X //

f

��

Y //

��

Z

��

// X[1]

f [1]

��X ′ // Y ′ // Z ′ // X ′[1]

in which both horizontal rows are distinguished triangles, there exists a dotted arrow rendering theentire diagram commutative.

(TR4) Suppose given three distinguished triangles

Xf→ Y u→ Y/X d→ X[1]

Yg→ Z v→ Z/Y d

′

→ Y [1]

Xg◦f→ Z w→ Z/X d

′′

→ X[1]in D. There exists a fourth distinguished triangle

Y/Xφ→ Z/X ψ→ Z/Y θ→ Y/X[1]

such that the diagram

Xg◦f //

f

��

Z

w

""

v // Z/Y

d′

""

θ // Y/X[1]

Y

u

!!

g

==

Z/X

ψ


Remark 1.1.2.6. If the ∞-category C is not clear from context, then we will denote the suspension andloop functors Σ,Ω : C→ C by ΣC and ΩC, respectively.

Notation 1.1.2.7. If C is a stable ∞-category and n ≥ 0, we let

X 7→ X[n]

denote the nth power of the suspension functor Σ : C→ C constructed above (this functor is well-defined upto canonical equivalence). If n ≤ 0, we let X 7→ X[n] denote the (−n)th power of the loop functor Ω. Wewill use the same notation to indicate the induced functors on the homotopy category hC.

Remark 1.1.2.8. If the ∞-category C is pointed but not necessarily stable, the suspension and loop spacefunctors need not be homotopy inverses but are nevertheless adjoint to one another (provided that bothfunctors are defined).

If C is a pointed ∞-category containing a pair of objects X and Y , then the space MapC(X,Y ) hasa natural base point, given by the zero map. Moreover, if C admits cofibers, then the suspension functorΣC : C→ C is essentially characterized by the existence of natural homotopy equivalences

MapC(Σ(X), Y )→ Ω MapC(X,Y ).

In particular, we conclude that π0 MapC(Σ(X), Y ) ' π1 MapC(X,Y ), so that π0 MapC(Σ(X), Y ) has thestructure of a group (here the fundamental group of MapC(X,Y ) is taken with base point given by thezero map). Similarly, π0 MapC(Σ

2(X), Y ) ' π2 MapC(X,Y ) has the structure of an abelian group. If thesuspension functor X 7→ Σ(X) is an equivalence of ∞-categories, then for every Z ∈ C we can choose Xsuch that Σ2(X) ' Z to deduce the existence of an abelian group structure on MapC(Z, Y ). It is easy to seethat this group structure depends functorially on Z, Y ∈ hC. We are therefore most of the way to provingthe following result:

Lemma 1.1.2.9. Let C be a pointed ∞-category which admits cofibers, and suppose that the suspensionfunctor Σ : C→ C is an equivalence. Then hC is an additive category.

Proof. The argument sketched above shows that hC is (canonically) enriched over the category of abeliangroups. It will therefore suffice to prove that hC admits finite coproducts. We will prove a slightly strongerstatement: the ∞-category C itself admits finite coproducts. Since C has an initial object, it will suffice totreat the case of pairwise coproducts. Let X,Y ∈ C, and let cofib : Fun(∆1,C) → C denote the functorwhich assign to each morphism its cofiber, so that we have equivalences X ' cofib(X[−1] u→ 0) and Y 'cofib(0

v→ Y ). Proposition T.5.1.2.2 implies that u and v admit a coproduct in Fun(∆1,C) (namely, the zeromap X[−1] 0→ Y ). Since the functor cofib preserves coproducts (Remark 1.1.1.8), we conclude that X andY admit a coproduct (which can be constructed as the cofiber of the zero map from X[−1] to Y ).

Let C be a pointed ∞-category which admits cofibers. By construction, any diagram

X //

��

0

��0′ // Y

which belongs to MΣ determines a canonical isomorphism X[1]→ Y in the homotopy category hC. We willneed the following observation:

Lemma 1.1.2.10. Let C be a pointed ∞-category which admits cofibers, and let

Xf //

f ′

��

0

��0′ // Y

1.1. FOUNDATIONS 21

be a diagram in C, classifying a morphism θ ∈ HomhC(X[1], Y ). (Here 0 and 0′ are zero objects of C.) Thenthe transposed diagram

Xf ′ //

f

��

0′

��0 // Y

classifies the morphism −θ ∈ HomhC(X[1], Y ). Here −θ denotes the inverse of θ with respect to the groupstructure on HomhC(X[1], Y ) ' π1 MapC(X,Y ).

Proof. Without loss of generality, we may suppose that 0 = 0′ and f = f ′. Let σ : Λ20 → C be the diagram

0f← X f→ 0.

For every diagram p : K → C, let D(p) denote the Kan complex Cp/×C{Y }. Then HomhC(X[1], Y ) 'π0 D(σ). We note that

D(σ) ' D(f)×D(X) D(f).

Since 0 is an initial object of C, D(f) is contractible. In particular, there exists a point q ∈ D(f). Let

D′ = D(f)×Fun({0},D(X)) Fun(∆1,D(X))×Fun({1},D(X)) D(f)

D′′ = {q} ×Fun({0},D(X)) Fun(∆1,D(X))×Fun({1},D(X)) {q}

so that we have canonical inclusions

D′′ ↪→ D′ ←↩ D(σ).

The left map is a homotopy equivalence because D(f) is contractible, and the right map is a homotopyequivalence because the projection D(f)→ D(X) is a Kan fibration. We observe that D′′ can be identifiedwith the simplicial loop space of HomLC(X,Y ) (taken with the base point determined by q, which we canidentify with the zero map from X to Y ). Each of the Kan complexes D(σ), D′, D′′ is equipped witha canonical involution. On D(σ), this involution corresponds to the transposition of diagrams as in thestatement of the lemma. On D′′, this involution corresponds to reversal of loops. The desired conclusion nowfollows from the observation that these involutions are compatible with the inclusions D′′,D(σ) ⊆ D′.

Definition 1.1.2.11. Let C be a pointed ∞-category which admits cofibers. Suppose given a diagram

Xf→ Y g→ Z h→ X[1]

in the homotopy category hC. We will say that this diagram is a distinguished triangle if there exists adiagram ∆1 ×∆2 → C as shown

Xf̃ //

��

Y

g̃

��

// 0

��0′ // Z

h̃ // W,

satisfying the following conditions:

(i) The objects 0, 0′ ∈ C are zero.

(ii) Both squares are pushout diagrams in C.

(iii) The morphisms f̃ and g̃ represent f and g, respectively.


(iv) The map h : Z → X[1] is the composition of (the homotopy class of) h̃ with the equivalence W ' X[1]determined by the outer rectangle.

Remark 1.1.2.12. We will generally only use Definition 1.1.2.11 in the case where C is a stable∞-category.However, it will be convenient to have the terminology available in the case where C is not yet known to bestable.

The following result is an immediate consequence of Lemma 1.1.2.10:

Lemma 1.1.2.13. Let C be a stable ∞-category. Suppose given a diagram ∆2 ×∆1 → C, depicted as

X

f

��

// 0

��Y

��

g // Z

h

��0′ // W,

where both squares are pushouts and the objects 0, 0′ ∈ C are zero. Then the diagram

Xf→ Y g→ Z −h

′

→ X[1]

is a distinguished triangle in hC, where h′ denotes the composition of h with the isomorphism W ' X[1] deter-mined by the outer square, and −h′ denotes the composition of h′ with the map − id ∈ HomhC(X[1], X[1]) 'π1 MapC(X,X[1]).

We can now state the main result of this section:

Theorem 1.1.2.14. Let C be a pointed ∞-category which admits cofibers, and suppose that the suspensionfunctor Σ is an equivalence. Then the translation functor of Notation 1.1.2.7 and the class of distinguishedtriangles of Definition 1.1.2.11 endow hC with the structure of a triangulated category.

Remark 1.1.2.15. The hypotheses of Theorem 1.1.2.14 hold whenever C is stable. In fact, the hypothesesof Theorem 1.1.2.14 are equivalent to the stability of C: see Corollary 1.4.2.27.

Proof. We must verify that Verdier’s axioms (TR1) through (TR4) are satisfied.

(TR1) Let E ⊆ Fun(∆1 ×∆2,C) be the full subcategory spanned by those diagrams

Xf //

��

Y

��

// 0

��0′ // Z // W

of the form considered in Definition 1.1.2.11, and let e : E → Fun(∆1,C) be the restriction to theupper left horizontal arrow. Repeated use of Proposition T.4.3.2.15 implies e is a trivial fibration. Inparticular, every morphism f : X → Y can be completed to a diagram belonging to E. This proves(a). Part (b) is obvious, and (c) follows from the observation that if f = idX , then the object Z in theabove diagram is a zero object of C.

(TR2) Suppose that

Xf→ Y g→ Z h→ X[1]

1.1. FOUNDATIONS 23

is a distinguished triangle in hC, corresponding to a diagram σ ∈ E as depicted above. Extend σ to adiagram

X //

��

Y

��

// 0

��0′ // Z //

��

W

u

��0′′ // V

where the lower right square is a pushout and 0′′ is a zero object of C. We have a map between thesquares

X //

��

0

��

Y //

��

0

��0′ // W 0′′ // V

which induces a commutative diagram in the homotopy category hC

W

u

��

// X[1]

f [1]

��V // Y [1]

where the horizontal arrows are isomorphisms. Applying Lemma 1.1.2.13 to the rectangle on the rightof the large diagram, we conclude that

Yg→ Z h→ X[1] −f [1]→ Y [1]

is a distinguished triangle in hC.

Conversely, suppose that

Yg→ Z h→ X[1] −f [1]→ Y [1]

is a distinguished triangle in hC. Since the functor Σ : C→ C is an equivalence, we conclude that thetriangle

Y [−2] g[−2]→ Z[−2] h[−2]→ X[−1] −f [−1]→ Y [−1]is distinguished. Applying the preceding argument five times, we conclude that the triangle

Xf→ Y g→ Z h→ X[1]

is distinguished, as desired.

(TR3) Suppose we are given distinguished triangles

Xf→ Y → Z → X[1]

X ′f ′→ Y ′ → Z ′ → X ′[1]

in hC. Without loss of generality, we may suppose that these triangles are induced by diagramsσ, σ′ ∈ E. Any commutative diagram

Xf //

��

Y

��X ′

f ′ // Y ′


in the homotopy category hC can be lifted (nonuniquely) to a square in C, which we may identifywith a morphism φ : e(σ) → e(σ′) in the ∞-category Fun(∆1,C). Since e is a trivial fibration ofsimplicial sets, φ can be lifted to a morphism σ → σ′ in E, which determines a natural transformationof distinguished triangles

X

��

// Y

��

// Z //

��

X[1]

��X ′ // Y ′ // Z ′ // X ′[1].

(TR4) Let f : X → Y and g : Y → Z be morphisms in C. In view of the fact that e : E → Fun(∆1,C) is atrivial fibration, any distinguished triangle in hC beginning with f , g, or g ◦ f is uniquely determinedup to (nonunique) isomorphism. Consequently, it will suffice to prove that there exist some triple ofdistinguished triangles which satisfies the conclusions of (TR4). To prove this, we construct a diagramin C

Xf //

��

Yg //

��

Z //

��

0

��0 // Y/X

��

// Z/X

��

// X ′ //

��

0

��0 // Z/Y // Y ′ // (Y/X)′

where 0 is a zero object of C, and each square in the diagram is a pushout (more precisely, we applyProposition T.4.3.2.15 repeatedly to construct a map from the nerve of the appropriate partially orderedset into C). Restricting to appropriate rectangles contained in the diagram, we obtain isomorphismsX ′ ' X[1], Y ′ ' Y [1], (Y/X)′ ' Y/X[1], and four distinguished triangles

Xf→ Y → Y/X → X[1]

Yg→ Z → Z/Y → Y [1]

Xg◦f→ Z → Z/X → X[1]

Y/X → Z/X → Z/Y → Y/X[1].

The commutativity in the homotopy category hC required by (TR4) follows from the (stronger) com-mutativity of the above diagram in C itself.

Remark 1.1.2.16. The definition of a stable ∞-category is quite a bit simpler than that of a triangulatedcategory. In particular, the octahedral axiom (TR4) is a consequence of ∞-categorical principles which arebasic and easily motivated.

Notation 1.1.2.17. Let C be a stable∞-category containing a pair of objects X and Y . We let ExtnC(X,Y )denote the abelian group HomhC(X[−n], Y ). If n is negative, this can be identified with the homotopygroup π−n MapC(X,Y ). More generally, Ext

nC(X,Y ) can be identified with the (−n)th homotopy group of

an appropriate spectrum of maps from X to Y .

1.1. FOUNDATIONS 25

1.1.3 Closure Properties of Stable ∞-CategoriesAccording to Definition 1.1.1.9, a pointed ∞-category C is stable if it admits certain pushout squares andcertain pullback squares, which are required to coincide with one another. Our goal in this section is toprove that a stable ∞-category C admits all finite limits and colimits, and that the pushout squares in Ccoincide with the pullback squares in general (Proposition 1.1.3.4). To prove this, we will need the followingeasy observation (which is quite useful in its own right):

Proposition 1.1.3.1. Let C be a stable ∞-category, and let K be a simplicial set. Then the ∞-categoryFun(K,C) is stable.

Proof. This follows immediately from the fact that fibers and cofibers in Fun(K,C) can be computed point-wise (Proposition T.5.1.2.2).

Definition 1.1.3.2. If C is stable ∞-category, and C0 is a full subcategory containing a zero object andstable under the formation of fibers and cofibers, then C0 is itself stable. In this case, we will say that C0 isa stable subcategory of C.

Lemma 1.1.3.3. Let C be a stable ∞-category, and let C′ ⊆ C be a full subcategory which is stable undercofibers and under translations. Then C′ is a stable subcategory of C.

Proof. It will suffice to show that C′ is stable under fibers. Let f : X → Y be a morphism in C. Theorem1.1.2.14 shows that there is a canonical equivalence fib(f) ' cofib(f)[−1].

Proposition 1.1.3.4. Let C be a pointed∞-category. Then C is stable if and only if the following conditionsare satisfied:

(1) The ∞-category C admits finite limits and colimits.

(2) A square

X //

��

Y

��X ′ // Y ′

in C is a pushout if and only if it is a pullback.

Proof. Condition (1) implies the existence of fibers and cofibers in C, and condition (2) implies that a trianglein C is a fiber sequence if and only if it is a cofiber sequence. This proves the “if” direction.

Suppose now that C is stable. We begin by proving (1). It will suffice to show that C admits finitecolimits; the dual argument will show that C admits finite limits as well. According to Proposition T.4.4.3.2,it will suffice to show that C admits coequalizers and finite coproducts. The existence of finite coproductswas established in Lemma 1.1.2.9. We now conclude by observing that a coequalizer for a diagram

Xf //f ′// Y

can be identified with cofib(f − f ′).We now show that every pushout square in C is a pullback; the converse will follow by a dual argument.

Let D ⊆ Fun(∆1 × ∆1,C) be the full subcategory spanned by the pullback squares. Then D is stableunder finite limits and under translations. It follows from Lemma 1.1.3.3 that D is a stable subcategory ofFun(∆1 ×∆1,C).

Let i : Λ20 ↪→ ∆1×∆1 be the inclusion, and let i! : Fun(Λ20,C)→ Fun(∆1×∆1,C) be a functor of left Kanextension. Then i! preserves finite colimits, and is therefore exact (Proposition 1.1.4.1). Let D

′ = i−1! D.


Then D′ is a stable subcategory of Fun(Λ20,C); we wish to show that D′ = Fun(Λ20,C). To prove this, we

observe that any diagramX ′ ← X → X ′′

can be obtained as a (finite) colimit

e′X′∐e′X

eX∐e′′X

e′′X′′

where eX ∈ Fun(Λ20,C) denotes the diagram X ← X → X, e′Z ∈ Fun(Λ20,C) denotes the diagram Z ← 0→ 0,and e′′Z ∈ Fun(Λ20,C) denotes the diagram 0 ← 0 → Z. It will therefore suffice to prove that a pushout ofany of these five diagrams is also a pullback. This follows immediately from the following more generalobservation: any pushout square

A //

f

��

A′

��B // B′

in an (arbitrary) ∞-category C is also a pullback square, provided that f is an equivalence.

Remark 1.1.3.5. Let C be a stable ∞-category. Then C admits finite products and finite coproducts(Proposition 1.1.3.4). Moreover, for any pair of objects X,Y ∈ C, there is a canonical equivalence

X q Y → X × Y,

given by the matrix [idX 00 idY

].

Theorem 1.1.2.14 implies that this map is an equivalence. We will sometimes use the notation X ⊕ Y todenote a product or coproduct of X and Y in C.

We conclude this section by establishing a few closure properties for the class of stable ∞-categories.

Proposition 1.1.3.6. Let C be a (small) stable ∞-category, and let κ be a regular cardinal. Then the∞-category Indκ(C) is stable.

Proof. The functor j preserves finite limits and colimits (Propositions T.5.1.3.2 and T.5.3.5.14). It followsthat j(0) is a zero object of Indκ(C), so that Indκ(C) is pointed.

We next show that every morphism f : X → Y in Indκ(C) admits a fiber and a cofiber. According toProposition T.5.3.5.15, we may assume that f is a κ-filtered colimit of morphisms fα : Xα → Yα whichbelong to the essential image C′ of j. Since j preserves fibers and cofibers, each of the maps fα has a fiberand a cofiber in Indκ. It follows immediately that f has a cofiber (which can be written as a colimit ofthe cofibers of the maps fα). The existence of fib(f) is slightly more difficult. Choose a κ-filtered diagramp : I→ Fun(∆1 ×∆1,C′), where each p(α) is a pullback square

Zα //

��

0

��Xα

fα // Yα.

Let σ be a colimit of the diagram p; we wish to show that σ is a pullback diagram in Indκ(C). Since Indκ(C)is stable under κ-small limits in P(C), it will suffice to show that σ is a pullback square in P(C). Since P(C)is an ∞-topos, filtered colimits in P(C) are left exact (Example T.7.3.4.7); it will therefore suffice to showthat each p(α) is a pullback diagram in P(C). This is obvious, since the inclusion C′ ⊆ P(C) preserves alllimits which exist in C′ (Proposition T.5.1.3.2).

1.1. FOUNDATIONS 27

To complete the proof, we must show that a triangle in Indκ(C) is a fiber sequence if and only if it is acofiber sequence. Suppose we are given a fiber sequence

Z //

��

0

��X // Y

in Indκ(C). The above argument shows that we can write this triangle as a filtered colimit of fiber sequences

Zα //

��

0

��Xα // Yα

in C′. Since C′ is stable, we conclude that these triangles are also cofiber sequences. The original triangle istherefore a filtered colimit of cofiber sequences in C′, hence a cofiber sequence. The converse follows by thesame argument.

Corollary 1.1.3.7. Let C be a stable ∞-category. Then the idempotent completion of C is also stable.Proof. According to Lemma T.5.4.2.4, we can identify the idempotent completion of C with a full subcategoryof Ind(C) which is closed under shifts and finite colimits.

1.1.4 Exact Functors

Let F : C → C′ be a functor between stable ∞-categories. Suppose that F carries zero objects into zeroobjects. It follows immediately that F carries triangles into triangles. If, in addition, F carries fibersequences to fiber sequences, then we will say that F is exact. The exactness of a functor F admits thefollowing alternative characterizations:

Proposition 1.1.4.1. Let F : C → C′ be a functor between stable ∞-categories. The following conditionsare equivalent:

(1) The functor F is left exact. That is, F commutes with finite limits.

(2) The functor F is right exact. That is, F commutes with finite colimits.

(3) The functor F is exact.

Proof. We will prove that (2) ⇔ (3); the equivalence (1) ⇔ (3) will follow by a dual argument. Theimplication (2)⇒ (3) is obvious. Conversely, suppose that F is exact. The proof of Proposition 1.1.3.4 showsthat F preserves coequalizers, and the proof of Lemma 1.1.2.9 shows that F preserves finite coproducts. Itfollows that F preserves all finite colimits (see the proof of Proposition T.4.4.3.2).

The identity functor from any stable ∞-category to itself is exact, and a composition of exact functors isexact. Consequently, there exists a subcategory CatEx∞ ⊆ Cat∞ in which the objects are stable ∞-categoriesand the morphisms are the exact functors. Our next few results concern the stability properties of thissubcategory.

Proposition 1.1.4.2. Suppose given a homotopy Cartesian diagram of ∞-categories

C′G′ //

F ′

��

C

F

��D′

G // D .

Suppose further that C, D′, and D are stable, and that the functors F and G are exact. Then:


(1) The ∞-category C′ is stable.

(2) The functors F ′ and G′ are exact.

(3) If E is a stable ∞-category, then a functor H : E→ C′ is exact if and only if the functors F ′ ◦H andG′ ◦H are exact.

Proof. Combine Proposition 1.1.3.4 with Lemma T.5.4.5.5.

Proposition 1.1.4.3. Let {Cα}α∈A be a collection of stable ∞-categories. Then the product

C =∏α∈A

Cα

is stable. Moreover, for any stable ∞-category D, a functor F : D → C is exact if and only if each of thecompositions

DF→ C→ Cα

is an exact functor.

Proof. This follows immediately from the fact that limits and colimits in C are computed pointwise.

Theorem 1.1.4.4. The ∞-category CatEx∞ admits small limits, and the inclusion

CatEx∞ ⊆ Cat∞

preserves small limits.

Proof. Using Propositions 1.1.4.2 and 1.1.4.3, one can repeat the argument used to prove PropositionT.5.4.7.3.

We have the following analogue of Theorem 1.1.4.4.

Proposition 1.1.4.5. Let p : X → S be an inner fibration of simplicial sets. Suppose that:

(i) For each vertex s of S, the fiber Xs = X ×S {s} is a stable ∞-category.

(ii) For every edge s→ s′ in S, the restriction X ×S ∆1 → ∆1 is a coCartesian fibration, associated to anexact functor Xs → Xs′ .

Then:

(1) The ∞-category MapS(S,X) of sections of p is stable.

(2) If C is an arbitrary stable ∞-category, and f : C → MapS(S,X) induces an exact functor Cf→

MapS(S,X)→ Xs for every vertex s of S, then f is exact.

(3) For every set E of edges of S, let Y (E) ⊆ MapS(S,X) be the full subcategory spanned by those sectionsf : S → X of p with the following property:

(∗) For every e ∈ E, f carries e to a pe-coCartesian edge of the fiber product X ×S ∆1, wherepe : X ×S ∆1 → ∆1 denotes the projection.

Then each Y (E) is a stable subcategory of MapS(S,X).

Proof. Combine Proposition T.5.4.7.11, Theorem 1.1.4.4, and Proposition 1.1.3.1.

Proposition 1.1.4.6. The ∞-category CatEx∞ admits small filtered colimits, and the inclusion CatEx∞ ⊆ Cat∞

preserves small filtered colimits.

1.2. STABLE ∞-CATEGORIES AND HOMOLOGICAL ALGEBRA 29

Proof. Let I be a filtered ∞-category, p : I→ CatEx∞ a diagram, which we will indicate by {CI}I∈I, and C acolimit of the induced diagram I→ Cat∞. We must prove:

(i) The ∞-category C is stable.

(ii) Each of the canonical functors θI : CI → C is exact.

(iii) Given an arbitrary stable ∞-category D, a functor f : C → D is exact if and only if each of thecomposite functors CI

θI→ C→ D is exact.

In view of Proposition 1.1.4.1, (ii) and (iii) follow immediately from Proposition T.5.5.7.11. The same resultimplies that C admits finite limits and colimits, and that each of the functors θI preserves finite limits andcolimits.

To prove that C has a zero object, we select an object I ∈ I. The functor CI → C preserves initial andfinal objects. Since CI has a zero object, so does C.

We will complete the proof by showing that every fiber sequence in C is a cofiber sequence (the conversefollows by the same argument). Fix a morphism f : X → Y in C. Without loss of generality, we may supposethat there exists I ∈ I and a morphism f̃ : X̃ → Ỹ in CI such that f = θI(f̃) (Proposition T.5.4.1.2). Forma pullback diagram σ̃

W̃ //

��

X̃

��0 // Ỹ

in CI . Since CI is stable, this diagram is also a pushout. It follows that θI(σ̃) is a triangle W → Xf→ Y

which is both a fiber sequence and a cofiber sequence in C.

1.2 Stable ∞-Categories and Homological AlgebraLet A be an abelian category with enough projective objects. In §1.3.2, we will explain how to associate to Aa stable ∞-category D−(A), whose objects are (right-bounded) chain complexes of projective objects of A.The homotopy category D−(A) is a triangulated category, which is usually called the derived category of A.We can recover A as a full subcategory of the triangulated category hD−(A) (or even as a full subcategory ofthe ∞-category D−(A)): namely, A is equivalent to the full subcategory spanned by those chain complexesPast satisfying Hn(P∗) ' 0 for n 6= 0. This subcategory can be described as the intersection

D−(A)≥0 ∩D−(A)≤0,

where D−(A)≤0 is defined to be the full subcategory spanned by those chain complexes P∗ with Hn(P∗) ' 0for n > 0, and D−(A)≥0 is spanned by those chain complexes with Hn(P∗) ' 0 for n < 0.

In §1.2.1, we will axiomatize the essence of the situation by reviewing the notion of a t-structure on astable ∞-category C. A t-structure on C is a pair of full subcategories (C≥0,C≤0) satisfying some axiomswhich reflect the idea that objects of C≥0 (C≤0) are “concentrated in nonnegative (nonpositive) degrees” (seeDefinition 1.2.1.1). In this case, one can show that the intersection C≥0 ∩C≤0 is equivalent to the nerve ofan abelian category, which we call the heart of C and denote by C♥. To any object X ∈ C, we can associatehomotopy objects πnX ∈ C♥ (in the special case C = D−(A), the functor πn associates to each chain complexP∗ its nth homology Hn(P∗)).

If C is a stable∞-category equipped with a t-structure, then it is often possible to relate questions aboutC to homological algebra in the abelian category C♥. In §1.2.2, we give an illustration of this principle, byshowing that every filtration on an object X ∈ C determines a spectral sequence {Ep,qr , dr}r≥1 in the abeliancategory C♥, which (in good cases) converges to the homotopy objects πnX (Proposition 1.2.2.7). The first


page of this spectral sequence has a reasonably explicit description in terms of the homotopy objects of thesuccessive quotients for the filtration of X. In practice, it is often difficult to describe Ep,qr when r > 2.However, there is a convenient description in the case r = 2, at least when X is given as the geometricrealization of a simplicial object X• of C (equipped with the corresponding skeletal filtration). In §1.2.4 wewill show that this is essentially no loss of generality: if C is a stable ∞-category, then every nonnegativelyfiltered object X of C can be realized as the geometric realization of a simplicial object of C, equipped withthe skeletal filtration (Theorem 1.2.4.1). This assertion can be regarded as an ∞-categorical analogue of theclassical Dold-Kan correspondence between simplicial objects and chain complexes in an abelian category,which we review in §1.2.3.

1.2.1 t-Structures on Stable ∞-CategoriesLet C be an ∞-category. Recall that we say a full subcategory C′ ⊆ C is a localization of C if the inclusionfunctor C′ ⊆ C has a left adjoint (§T.5.2.7). In this section, we will introduce a special class of localizations,called t-localizations, in the case where C is stable. We will further show that there is a bijective correspon-dence between t-localizations of C and t-structures on the triangulated category hC. We begin with a reviewof the classical theory of t-structures; for a more thorough introduction we refer the reader to [13].

Definition 1.2.1.1. Let D be a triangulated category. A t-structure on D is defined to be a pair of fullsubcategories D≥0, D≤0 (always assumed to be stable under isomorphism) having the following properties:

(1) For X ∈ D≥0 and Y ∈ D≤0, we have HomD(X,Y [−1]) = 0.

(2) We have inclusions D≥0[1] ⊆ D≥0, D≤0[−1] ⊆ D≤0.

(3) For any X ∈ D, there exists a fiber sequence X ′ → X → X ′′ where X ′ ∈ D≥0 and X ′′ ∈ D≤0[−1].

Notation 1.2.1.2. If D is a triangulated category equipped with a t-structure, we will write D≥n for D≥0[n]and D≤n for D≤0[n]. Observe that we use a homological indexing convention.

Remark 1.2.1.3. In Definition 1.2.1.1, either of the full subcategories D≥0,D≤0 ⊆ D determines the other.For example, an object X ∈ D belongs to D≤−1 if and only if HomD(Y,X) vanishes for all Y ∈ D≥0.

Definition 1.2.1.4. Let C be a stable ∞-category. A t-structure on C is a t-structure on the homotopycategory hC. If C is equipped with a t-structure, we let C≥n and C≤n denote the full subcategories of Cspanned by those objects which belong to (hC)≥n and (hC)≤n , respectively.

Proposition 1.2.1.5. Let C be a stable ∞-category equipped with a t-structure. For each n ∈ Z, the fullsubcategory C≤n is a localization of C.

Proof. Without loss of generality, we may suppose n = −1. According to Proposition T.5.2.7.8, it will sufficeto prove that for each X ∈ C, there exists a map f : X → X ′′, where X ′′ ∈ C≤−1 and for each Y ∈ C≤−1,the map

MapC(X′′, Y )→ MapC(X,Y )

is a weak homotopy equivalence. Invoking part (3) of Definition 1.2.1.1, we can choose f to fit into a fibersequence

X ′ → X f→ X ′′

where X ′ ∈ C≥0. According to Whitehead’s theorem, we need to show that for every k ≤ 0, the map

ExtkC(X′′, Y )→ ExtkC(X,Y )

is an isomorphism of abelian groups. Using the long exact sequence associated to the fiber sequence above,we are reduced to proving that the groups ExtkC(X

′, Y ) vanish for k ≤ 0. We now use condition (2) ofDefinition 1.2.1.1 to conclude that X ′[−k] ∈ C≥0. Condition (1) of Definition 1.2.1.1 now implies that

ExtkC(X′, Y ) ' HomhC(X ′[−k], Y ) ' 0.

1.2. STABLE ∞-CATEGORIES AND HOMOLOGICAL ALGEBRA 31

Corollary 1.2.1.6. Let C be a stable ∞-category equipped with a t-structure. The full subcategories C≤n ⊆ Care stable under all limits which exist in C. Dually, the full subcategories C≥n ⊆ C are stable under all colimitswhich exist in C.

Notation 1.2.1.7. Let C be a stable ∞-category equipped with a t-structure. We will let τ≤n denote a leftadjoint to the inclusion C≤n ⊆ C, and τ≥n a right adjoint to the inclusion C≥n ⊆ C.

Remark 1.2.1.8. Fix n,m ∈ Z, and let C be a stable ∞-category equipped with a t-structure. Then thetruncation functors τ≤n, τ≥n map the full subcategory C≤m to itself. To prove this, we first observe thatτ≤n is equivalent to the identity on C≤m if m ≤ n, while if m ≥ n the essential image of τ≤n is contained inC≤n ⊆ C≤m. To prove the analogous result for τ≥n, we observe that the proof of Proposition 1.2.1.5 impliesthat for each X, we have a fiber sequence

τ≥nX → Xf→ τ≤n−1X.

If X ∈ C≤m, then τ≤n−1X also belongs to C≤m, so that τ≥nX ' fib(f) belongs to C≤m since C≤m is stableunder limits.

Warning 1.2.1.9. In §T.5.5.6, we introduced for every∞-category C a full subcategory τ≤n C of n-truncatedobjects of C. In that context, we used the symbol τ≤n to denote a left adjoint to the inclusion τ≤n C ⊆ C.This is not compatible with Notation 1.2.1.7. In fact, if C is a stable ∞-category, then it has no nonzerotruncated objects at all: if X ∈ C is nonzero, then the identity map from X to itself determines a nontrivialhomotopy class in πn MapC(X[−n], X), for all n ≥ 0. Nevertheless, the two notations are consistent whenrestricted to C≥0, in view of the following fact:

• Let C be a stable ∞-category equipped with a t-structure. An object X ∈ C≥0 is k-truncated (as anobject of C≥0) if and only if X ∈ C≤k.

In fact, we have the following more general statement: for any X ∈ C and k ≥ −1, X belongs to C≤k ifand only if MapC(Y,X) is k-truncated for every Y ∈ C≥0. Because the latter condition is equivalent to thevanishing of ExtnC(Y,X) for n < −k, we can use the shift functor to reduce to the case where n = 0 andk = −1, which is addressed by Remark 1.2.1.3.

Let C be a stable ∞-category equipped with a t-structure, and let n,m ∈ Z. Remark 1.2.1.8 implies thatwe have a commutative diagram of simplicial sets

C≥n //

τ≤m

��

C

τ≤m

��C≥n ∩C≤m // C≤m .

As explained in §T.7.3.1, we get an induced transformation of functors

θ : τ≤m ◦ τ≥n → τ≥n

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